Codes for the Z-Channel

Nikita Polyanskii, Yihan Zhang

Abstract: This paper is a collection of results on combinatorial properties of codes for the Z-channel. A Z-channel with error fraction $\tau $ takes as input a length- $n$ binary codeword and injects in an adversarial manner up to $n\tau $ asymmetric errors, i.e., errors that only zero out bits but do not flip 0’s to 1’s. It is known that the largest $(L-1)$ -list-decodable code for the Z-channel with error fraction $\tau $ has exponential size (in $n$ ) if $\tau $ is less than a critical value that we call the $(L-1)$ -list-decoding Plotkin point and has constant size if $\tau $ is larger than the threshold. The $(L-1)$ -list-decoding Plotkin point is known to be $L^{-({1}/{L-1})} - L^{-({L}/{L-1})} $ , which equals 1/4 for unique-decoding with $L-1=1 $ . In this paper, we derive various results for the size of the largest codes above and below the list-decoding Plotkin point. In particular, we show that the largest $(L-1)$ -list-decodable code $\varepsilon $ -above the Plotkin point, for any given sufficiently small positive constant $\varepsilon >0 $ , has size $\Theta _{L}(\varepsilon ^{-3/2})$ for any $L-1\ge 1$ . We also devise upper and lower bounds on the exponential size of codes below the list-decoding Plotkin point.